THE TURN
One might be asking how the lattice relates to the actual individual narratives beyond “There and Back Again”. We don’t have space here to say much. To paraphrase Waterfield’s translation of Plato’s The Republic:
“Then, my noble friend, geometry will draw the soul towards truth, and create the spirit of philosophy, and raise up that which is now unhappily allowed to fall down” (Waterfield 10.258).
I have good additional evidence to present elsewhere that, rather like Dante’s Divine Comedy, each of Tolkien’s narratives is a spiritual journey in a symbolic landscape, upwards through rational planes towards Truth. Each crossing through the Door requires a Turn. Tolkien uses a device, the TURN[13], which itself consists of three phases, three turns,—consider Frodo’s remark: “The third turn may turn the best” (LotR 4.III.647). Each TURN is 90 degrees as we see in Afterwards. If we trace three turns of the same handedness we begin to see a spiral. This TURN is laid out, that is, the Platonic ideal form is created, in the Music of the Ainur at the discords of Melkor in Ilúvatar’s hand signs as A, B and C. In my opinion, Williams is clearly aware of this system which is inspired by Platonism. His poem For a Picture encapsulates much of Tolkien’s grammar:
“OF THE MADONNA teaching the Divine Child geometry by figures drawn on the sand (Merejkowski)
STRAIGHT lines and circles, triangles and squares,
With exact finger on the sand she draws,
Marking how each new truth from its own cause
Demonstrably arises, and prepares
Place for more truth to follow”
[…]
The union of dimension, ideal line,
Inclusive number, and the circle squared” (Windows of Night 129).
Regarding the last two lines of the poem, in the heraldry he devised, Tolkien told us that women are circles, men are squares: “women within a circle personal Men within a lozenge [personal]” (A&I 191). Looking at the heraldic device “Lúthien II” (Pictures 47) for his wife Edith, when the curved lines are extended outside the bounds of the device, they imply the same geometry required to create the “Lunes of Hippocrates” (File:Lune of Hippocrates), a proposed geometric solution to the squaring of the circle.
Grownupishness (A&I 39) depicts a scene in which squares and circles are visually fighting accompanied by the words “Sightless Blind Well Wrapped-up”. “Wrap”: “bind up, swaddle […] from PIE *werp- “to turn, wind,” from root *wer- (2) “to turn, bend“. “Wife”: “Some proposed PIE roots for wife include *weip- “to twist, turn, wrap,” perhaps with sense of “veiled person”“. H&S tell us “Tolkien had himself grown up on his twenty-first birthday only a few months before he made the drawing” (A&I 38). We remember Bilbo’s words to Frodo:
“It’s a dangerous business, Frodo, going out of your door,” he used to say. “You step into the Road, and if you don’t keep your feet, there is no knowing where you might be swept off to” (LotR 1.III.74).
Frodo at his departure says “I am being swept off my feet at last” (LotR 1.I.35). In Grownupishness Tolkien has been swept off his feet. If we look at “sweep”, we can link this to Williams’s and Plato’s words: ““hasten, rush, move or pass along swiftly and strongly” […] “drive quickly, impel, move or carry forward by force.” […] “draw or drag something over”“. That is, “drawing the soul towards Truth” and “With exact finger on the sand she draws, Marking how each new truth”. Bilbo, like Frodo has to go out of his Door at Bag-End. Thus we can identify Grownupishness to be the moment of going through the Door, that is the moment Ilúvatar raises both of his hands in a command,— we note the hands of the figure in Grownupishness are also both outstretched. This is supported by the finger on the Hobbit map pointing to the Magic Door, which I suggest is the same finger drawing in the sand. If when we pass through the Door we rise to higher rational planes to higher Truths, this agrees with Williams:
“each new truth from its own cause
Demonstrably arises, and prepares
Place for more truth to follow”.
I propose that this is the same “following” found in the walking poem “down from the door where it began […] and I must follow if I can, Pursuing it with eager feet” (LotR 1.I.35). If Men are “the Followers” (Silm. 19), and Edith is Lúthien an elf, this makes sense. This analysis compliments our observations regarding their quarrel, on the domineering “kvan-riki”, and my suggested pun on the “quarrel” (square), “row”, the “lozenge”. In Grownupishness, the circles dominate the squares. Tolkien no doubt felt the pressure of being pushed into the unknown to make a viable life for himself and his wife. His translation of “I Love Sixpence” (J.R.R. Tolkien, E.V. Gordon & others 7) continues the theme of the put-upon husband in the “happy war”.
The author has compiled a lot of evidence that in LotR the largest scale structure is a set of TURNS centred on the Tongue of Lóthlorien. At this scale we have a large “X” centred over the map with the Tongue at the intersection. The Turns of the “wheels” (mentioned in the Council of Elrond) are centred on the Door which is the plane of a mirror. Note, just as a TURN is a set of three turns, Tolkien’s system creates wheels (turns) within wheels (turns). It’s my firm belief that Tolkien used his dialectic in every one of his creative works, —compare his character Father Christmas raising both of his hands in Letters from 1925 (Letters from Father Christmas 20) and 1926 (29) with Ilúvatar’s hand sequence. Tolkien’s statement about Hobbit being influenced by Carroll’s Through the Looking-Glass (TtLG),— “much closer in every way” to Lewis Carroll’s “Looking-glass” (Letters 26, #15), equally applies to LotR. The chessboard which is found in Looking-Glass is echoed in the floor of his picture The Hall of Bag-End, Residence of B.Baggins Esquire” (Pictures 20), which also has two looking-glasses on the east and west walls.
In the east mirror, if we look carefully, we see a diagonal ray of light. The door of Bag-End is flanked again by two trees which turns around the hill in exactly the same arrangement as Before and Afterwards, and Eeriness. In Tolkien’s picture The Forest of Lothlórien in Spring (Pictures 25), fig. 1.7 below, we can see a path which turns, a diagonal ray of sunlight, and if we look carefully we can see that the path leads to an arched door, with its suggestion of wood panelling. The Door suggested by interwoven branches repeats Tumble Hill and Trials for Doors of Durin decorations. The diagonal indicates the “Straight Road”, C, the plane of the hypotenuse as visible during the day or night (as previously stated), which the wizard is on in Eeriness. We can also see the suggestion of the Two Trees and 
Sun and Moon in the tree forms to either side of the path. If Edith is the Sun, this suggests that the diagonal ray of light in the Lothlórien picture is the finger which shows Tolkien the way to the Door and draws him onwards and upwards,—that keeps him on the “straight and narrow” if he “listens”. And we’ve just identified this moment of passing through the Door with C which is the hypotenuse, the diagonal. The ray of light visible in the east mirror of Bag-End and the details echoing Eeriness et al, further support our identification of the ray of light with the Sun drawing the soul through the Door to higher Truths. And we can surely identify this ray of light as the eucatastrophic “happy turn”, the “sudden glimpse of the truth behind the apparent Anankê […] of our world, […] a glimpse that is actually a ray of light through the very chinks of the universe about us”. This agrees with our hypothesis of the rotating World which began turning with Melkor’s discords, and that the ray of light during the day or night is a reminder, a pointer to the spiritually attentive. Moreover this ray is the “one chord […] piercing as the light of the eye of Ilúvatar” (Silm. 17). “Chord” is: “a shortening of accord“. Accord: “agree, be in harmony,” […] literally “be of one heart, bring heart to heart,” PIE root *kerd- “heart”“. It shows the way to the Magic Door, the heart at the centre of Eeriness. This is supported by the “X” shaped “single flowering rune” adorning the box given to Sam by Galadriel at the Tongue (TAotLotR 65-66). The “※” figure can also be found on page 66. Ilúvatar at the moment of maximum discord (the two trees in Eeriness on the “outside”), the two musics “utterly at variance”, draws the two through the Door back to the centre, but on a higher rational plane[14]. This grammar, the two interweaving trees forming a door, and the ray of light through the chinks can be found once again in Hobbit. At the entrance to Mirkwood we find:
“They walked in single file. The entrance to the path was like a sort of arch leading into a gloomy tunnel made by two great trees that leant together, too old and strangled with ivy and hung with lichen to bear more than a few blackened leaves. The path itself was narrow and wound in and out among the trunks. Soon the light at the gate was like a little bright hole far behind” […] all the trees leaned over them and listened. […] Occasionally a slender beam of sun that had the luck to slip in through some opening in the leaves far above, and still more luck in not being caught in the tangled boughs and matted twigs beneath, stabbed down thin and bright before them” (Hobbit 8.117).
We also recall the butterflies Bilbo sees on the top of the tree canopy. In the closing lines of LotR we see all of the elements present for the TURN, the turn, the ray of light, and the drawing through the door, and the star (Elanor the “Sun Star”):
“But Sam turned to Bywater, and so came back up the Hill, as day was ending once more. And he went on, and there was yellow light, and fire within; and the evening meal was ready, and he was expected. And Rose drew him in, and set him in his chair, and put little Elanor upon his lap.
He drew a deep breath. ‘Well, I’m back,’ he said” (LotR 6.IX.1031).
Sam has passed “back” through the Door of “There and Back Again”.
[13] I refer to the turn we see in Afterwards as the TURN. This itself is comprised of three turns, which can be regarded as the process of travelling down the corridor we see in Before towards the Door. This process can be viewed as the steps required for repentance: turns of spirit, physical reorientation, language utterance (Confession). At the time of writing this is the author’s understanding.
[14] A visual illustration of the processes in the Music of the Ainur, using the lattice, must wait for another time.
